# An Introduction to Elliptic Curves

Elliptic Curves and the Abelian Grou

## An Introduction to Elliptic Curves

Elliptic Curves are curves of the form
$y^2=x^3+ax+b$
This is called a Weierstrass equation. The rational points on this elliptic curve miraculously create an abelian group under an operation called point addition.

Given two points
$P$
and
$Q$
, we can calculate a third point
$P \oplus Q$
on the elliptic curve itself using a simple algorithm:
The images here are from Joseph Silverman's presentation slides from 2006, which are an excellent reference and can be found here.
First, we draw the line
$L$
through the points
$P$
and
$Q$
, which intersects the curve at a third point
$R$
. We then draw a vertical line passing through
$R$
which intersects the curve at a second point we call
$R^\prime$
.
$R^\prime$
is the result of
$P \oplus Q$
$P$
and
$Q$
.

### Adding a Point to Itself

We can also add a point
$P$
to itself, but how can we do that with infinitely many lines passing through? We simply take the tangent to the curve and find the point
$R$
it intersects at before mirroring the point to
$R^\prime = 2P$
.

### The Point at Infinity

Rarely, you may find yourself adding
$P$
to
$-P$
, the point directly below
$P$
. In this case there is no third point of intersection and we say the result of point addition is
$\mathcal{O}$
, the point "at infinity".